not contribute to the dynamics, so it may be neglected, giving: Next, define the so-called number operator by: . Consider the commutator of this operator with the atom-field Hamiltonian: Thus the number operator commutes with the atom-field Hamiltonian. The eigenstates of the number operator are the basis of tensor product states where the states of the field are the those with a definite number of photons. The number operator counts the total number of quanta in the atom-field system. In this basis of eigenstates of (total number states), the Hamiltonian takes on a block diagonal structure: With the exception of the scalar , each on the diagonal is itself a matrix of the form; Now, using the relation: We obtain the portion of the Hamiltonian that acts in the nth subspace as: By shifting the energy from to with the amount of , we can get where we have identified as the Rabi frequency of the system, and is the so-called "detuning" between the frequencies of the cavity and atomic transition. We have also defined the operators: to be the identity operator and Pauli x and z operators in the Hilbert space of the nth energy level of the atom-field system. This simple Hamiltonian is of the same form as what would be found in the Rabi problem. Diagonalization gives the energy eigenvalues and eigenstates to be: Where the angle is defined by the relation . Vacuum Rabi oscillations Consider an atom entering the cavity initially in its excited state, while the cavity is initially in the vacuum state. Then the state of the atom-field system as a function of time is: So the probabilities to find the system in the ground or excited states after interacting with the cavity for a time are: Thus the probability amplitude to find the atom in either state oscillates. This is the quantum mechanical explanation for the phenomenon of vacuum Rabi oscillation. In this case, there was only a single quantum in the atom-field system, carried in by the initially excited atom. In general, the Rabi oscillation associated with an atom-field system of quanta will have frequency . As explained below, this discrete spectrum of frequencies is the underlying reason for the collapses and subsequent revivals probabilities in the model. Jaynes-Cummings ladder As shown in the previous subsection, if the initial state of the atom-cavity system is or , as is the case